Abstract Algebra Teaching and Learning

Thomas Hausberger (1)

1 IMAG - Institut Montpelliérain Alexander Grothendieck

Algebra, also called modern algebra or structural algebra, denotes, by opposition to classical algebra which focuses on formal manipulation of abstract symbols in relation to solving equations, the new image of algebra that emerged at the turn of the 20 th century. It was first presented in history in the textbook Moderne Algebra (Waerden 1930) which "inverted the conceptual hierarchy of classical algebra" (Corry 2016) by putting algebraic structures to the fore and seeing properties of numbers, polynomials, etc. as consequences of what is known on general structures rather than the other way round. In an educational context, Abstract Algebra usually designates an upper division undergraduate course typically required for mathematics majors and centered on the structures of groups, rings, and fields. In general, it is encountered by students upon completion of courses in Linear Algebra. Abstract Algebra courses are thus taught at the transition between undergraduate and graduate studies and range from introductory to more advanced courses related to Galois Theory, Algebraic Number Theory, Algebraic Geometry, Homological Algebra or even Category Theory as a general theory of structures.

Keywords: Algebra; Modern Algebra; Group Theory; Ring Theory; Field Theory; Algebraic structures; Mathematical structuralism; Axiomatic method; Conceptual mathematics; Mathematical abstraction; University Mathematics Education

Domains: History and Overview [math.HO]; Education

Thomas Hausberger. Abstract Algebra Teaching and Learning. Steve Lerman. Encyclopedia of Mathematics Education, Springer, 2018, 978-94-007-4977-1. ⟨10.1007/978-3-319-77487-9_100022-1⟩. ⟨hal-01865229⟩